Topics
Rotational Dynamics
 Rotational Dynamics
 Circular Motion and Its Characteristics
 Applications of Uniform Circular Motion
 Vertical Circular Motion
 Moment of Inertia as an Analogous Quantity for Mass
 Radius of Gyration
 Theorems of Perpendicular and Parallel Axes
 Angular Momentum or Moment of Linear Momentum
 Expression for Torque in Terms of Moment of Inertia
 Conservation of Angular Momentum
 Rolling Motion
Circular Motion
 Angular Displacement
 Angular Velocity
 Angular Acceleration
 Angular Velocity and Its Relation with Linear Velocity
 Uniform Circular Motion
 Radial Acceleration
 Dynamics of Uniform Circular Motion  Centripetal Force
 Centrifugal Forces
 Banking of Roads
 Vertical Circular Motion Due to Earth’s Gravitation
 Equation for Velocity and Energy at Different Positions of Vertical Circular Motion
 Kinematical Equations for Circular Motion in Analogy with Linear Motion.
Gravitation
 Newton’s Law of Gravitation
 Projection of Satellite
 Periodic Time
 Kepler’s Laws
 Binding Energy and Escape Velocity of a Satellite
 Weightlessness
 Variation of ‘G’ Due to Lattitude and Motion
 Acceleration Due to Gravity and Its Variation with Altitude and Depth
 Communication satellite and its uses
 Composition of Two S.H.M.’S Having Same Period and Along Same Line
Mechanical Properties of Fluids
 Fluid and Its Properties
 Thrust and Pressure
 Liquid Pressure
 Pressure Exerted by a Liquid Column
 Atmospheric Pressure
 Gauge Pressure and Absolute Pressure
 Hydrostatic Paradox
 Transmission of Pressure in Liquids: Pascal’s Law
 Application of Pascal’s Law
 Measurement of Atmospheric Pressure
 Mercury Barometer (Simple Barometer)
 Open Tube Manometer
 Surface Tension
 Molecular Theory of Surface Tension
 Surface Tension and Surface Energy
 Angle of Contact
 Effect of Impurity and Temperature on Surface Tension
 Excess Pressure Across the Free Surface of a Liquid
 Explanation of Formation of Drops and Bubbles
 Capillarity and Capillary Action
 Fluids in Motion
 Critical Velocity and Reynolds Number
 Viscous Force Or Viscosity
 Stokes’ Law
 Terminal Velocity
 Equation of Continuity
 Bernoulli's Equation
 Applications of Bernoulli’s Equation
Angular Momentum
Kinetic Theory of Gases and Radiation
 Gases and Its Characteristics
 Classification of Gases: Real Gases and Ideal Gases
 Mean Free Path
 Pressure of Ideal Gas
 Root Mean Square (RMS) Speed
 Interpretation of Temperature in Kinetic Theory
 Law of Equipartition of Energy
 Specific Heat Capacity
 Absorption, Reflection, and Transmission of Heat Radiation
 Perfect Blackbody
 Emission of Heat Radiation
 Kirchhoff’s Law of Heat Radiation and Its Theoretical Proof
 Spectral Distribution of Blackbody Radiation
 Wien’s Displacement Law
 Stefanboltzmann Law of Radiation
Oscillations
 Periodic and Oscillatory Motions
 Simple Harmonic Motion (SHM)
 Differential Equation of Linear S.H.M.
 Projection of U.C.M.(Uniform Circular Motion) on Any Diameter
 Phase of K.E (Kinetic Energy)
 K.E.(Kinetic Energy) and P.E.(Potential Energy) in S.H.M.
 Composition of Two S.H.M.’S Having Same Period and Along Same Line
 Some Systems Executing Simple Harmonic Motion
Thermodynamics
Oscillations
 Oscillations
 Explanation of Periodic Motion
 Linear Simple Harmonic Motion (S.H.M.)
 Differential Equation of Linear S.H.M.
 Acceleration (a), Velocity (v) and Displacement (x) of S.H.M.
 Amplitude (A), Period (T) and Frequency (N) of S.H.M.
 Reference Circle Method
 Phase in S.H.M.
 Graphical Representation of S.H.M.
 Composition of Two S.H.M.’S Having Same Period and Along Same Line
 The Energy of a Particle Performing S.H.M.
 Simple Pendulum
 Angular S.H.M. and It's Differential Equation
 Damped Oscillations
 Free Oscillations, Forced Oscillations and Resonance Oscillations
 Periodic and Oscillatory Motions
Elasticity
Surface Tension
Superposition of Waves
Wave Motion
Wave Optics
Stationary Waves
Electrostatics
 Electrostatics
 Application of Gauss' Law
 Electric Potential and Potential Energy
 Electric Potential Due to a Point Charge, a Dipole and a System of Charges
 Equipotential Surfaces
 Electrical Energy of Two Point Charges and of a Dipole in an Electrostatic Field
 Conductors and Insulators, Free Charges and Bound Charges Inside a Conductor
 Dielectrics and Electric Polarisation
 Capacitors and Capacitance, Combination of Capacitors in Series and Parallel
 Displacement Current
 Energy Stored in a Capacitor
 Van De Graaff Generator
 Uniformly Charged Infinite Plane Sheet and Uniformly Charged Thin Spherical Shell (Field Inside and Outside)
Current Electricity
Kinetic Theory of Gases and Radiation
 Concept of an Ideal Gas
 Kinetic Theory of Gases Assumptions
 Mean Free Path
 Derivation for Pressure of a Gas
 Degrees of Freedom
 Derivation of Boyle’s Law
 Thermal Equilibrium
 First Law of Thermodynamics
 Heat Engines
 Heat and Temperature
 Qualitative Ideas of Blackbody Radiation
 Wein'S Displacement Law
 Green House Effect
 Stefan's Law
 Maxwell Distribution
 Specific Heat Capacities  Gases
 Law of Equipartition of Energy
Magnetic Fields Due to Electric Current
 Magnetic Fields Due to Electric Current
 Magnetic Force
 Cyclotron Motion
 Helical Motion
 Magnetic Force on a Wire Carrying a Current
 Force on a Closed Circuit in a Magnetic Field
 Torque on a Current Loop in Magnetic Field
 Magnetic Dipole Moment
 Magnetic Potential Energy of a Dipole
 Magnetic Field Due to a Current: Biotsavart Law
 Force of Attraction Between Two Long Parallel Wires
 Magnetic Field Produced by a Current in a Circular Arc of a Wire
 Axial Magnetic Field Produced by Current in a Circular Loop
 Magnetic Lines for a Current Loop
 Ampere's Law
 Magnetic Field of a Solenoid and a Toroid
Wave Theory of Light
Magnetic Materials
Interference and Diffraction
 Interference of Light
 Conditions for Producing Steady Interference Pattern
 Interference of Light Waves and Young’S Experiment
 Analytical Treatment of Interference Bands
 Measurement of Wavelength by Biprism Experiment
 Fraunhofer Diffraction Due to a Single Slit
 Rayleigh’s Criterion
 Resolving Power of a Microscope and Telescope
 Difference Between Interference and Diffraction
Electromagnetic Induction
 Electromagnetic Induction
 Faraday's Laws of Electromagnetic Induction
 Lenz's Law
 Flux of the Field
 Motional Electromotive Force
 Induced Emf in a Stationary Coil in a Changing Magnetic Field
 Generators
 Back Emf and Back Torque
 Induction and Energy Transfer
 Eddy Currents
 SelfInductance
 Energy Stored in a Magnetic Field
 Energy Density of a Magnetic Field
 Mutual Inductance
 Transformers
Electrostatics
 Applications of Gauss’s Law
 Mechanical Force on Unit Area of a Charged Conductor
 Energy Density of a Medium
 Dielectrics and Polarisation
 Concept of Condenser
 The Parallel Plate Capacitor
 Capacity of Parallel Plate Condenser
 Effect of Dielectric on Capacity
 Energy of Charged Condenser
 Condensers in Series and Parallel,
 VandeGraaff Generator
AC Circuits
Current Electricity
Dual Nature of Radiation and Matter
Magnetic Effects of Electric Current
Structure of Atoms and Nuclei
Magnetism
Semiconductor Devices
Electromagnetic Inductions
 Electromagnetic Induction
 Faraday’s Law of Induction
 SelfInductance
 Mutual Inductance
 Transformers
 Need for Displacement Current
 Coil Rotating in Uniform Magnetic Induction
 Alternating Currents
 Reactance and Impedance
 LC Oscillations
 Inductance and Capacitance
 Resonant Circuit
 Power in Ac Circuit: the Power Factor
 Lenz’S Law and Conservation of Energy
Electrons and Photons
Atoms, Molecules and Nuclei
 Alphaparticle Scattering and Rutherford’S Nuclear Model of Atom
 Bohr’s Model for Hydrogen Atom
 Hydrogen Spectrum
 Atomic Masses and Composition of Nucleus
 Introduction of Radioactivity
 Law of Radioactive Decay
 MassEnergy Relation and Mass Defect
 Nuclear Binding Energy
 Nuclear Fusion – Energy Generation in Stars
 deBroglie Relation
 Wave Nature of Matter
 Wavelength of an Electron
 DavissonGermer Experiment
 Continuous and Characteristics Xrays
Semiconductors
 Energy Bands in Solids
 Extrinsic Semiconductor
 Applications of Ntype and Ptype Semiconductors
 Special Purpose Pn Junction Diodes
 Semiconductor Diode
 Zener Diode as a Voltage Regulator
 IV Characteristics of Led
 Transistor and Characteristics of a Transistor
 Transistor as an Amplifier (Ceconfiguration)
 Transistor as a Switch
 Oscillators
 Digital Electronics and Logic Gates
Communication Systems
 Elements of a Communication System
 Basic Terminology Used in Electronic Communication Systems
 Bandwidth of Signals
 Bandwidth of Transmission Medium
 Need for Modulation and Demodulation
 Production and Detection of an Amplitude Modulated Wave
 Space Communication
 Propagation of Electromagnetic Waves
 Modulation and Its Necessity
notes
Surface Energy

Surface energy is the excess energy exhibited by the liquid molecules on the surface compared to those inside the liquid.

This means liquid molecules at the surface have greater energy as compared to molecules inside it.

Suppose there is a tumbler and when we pour water in the tumbler, it takes the shape of the tumbler.

It acquires free surface.
Case 1: When molecules are inside the liquid:

Suppose there is a molecule inside the water, there will be several other molecules that will attract that molecule in all the directions.

As a result this attraction will bind all the molecules together.

This results in negative potential energy of the molecule as it binds the molecule.

To separate this molecule huge amount of energy is required to overcome potential energy.

Some external energy is required to move this molecule and it should be greater than the potential energy.

Therefore in order to separate this molecule a huge amount of energy is required.
Case2: When the molecules are at the surface:
 When the molecule is at the surface, half of it will be inside and half of it is exposed to the atmosphere.

For the lower half of the molecule it will be attracted by the other molecules inside the liquid.

But the upper half is free. The negative potential energy is only because of lower half.

But the magnitude is half as compared to the potential energy of the molecule which is fully inside the liquid.

So the molecule has some excess energy, because of this additional energy which the molecules have at the surface different phenomenon happen like surface energy, surface tension.

Liquids always tend to have least surface are when left to itself.

As more surface area will require more energy as a result liquids tend to have least surface area.
Surface energy for two fluids in contact:

Whenever there are two fluids, in contact, surface energy depends on materials of the surfaces in contact.

Surface energy decreases if the molecules of the two fluids attract.

Surface energy increases if molecules of the two fluids repel.
Surface Tension

Surface tension is the property of the liquid surface which arises due to the fact that surface molecules have extra energy.

Surface energy is the extra energy that the molecules at the surface have.

Surface tension is the property of the liquid surface because the molecules have extra energy.

Surface energy is defined as surface energy per unit area of the liquid surface.

Denoted by ’S’.
Mathematically:

Consider a case in which liquid is enclosed in a movable bar.

Slide the bar slightly and it moves some distance (‘d’).

There will be increase in the area, (dl) where l=length of the bar.

Liquids have two surfaces one on the bar and others above the bar. Therefore area=2(dl)

Work done for this change =F x displacement.

Surface tension(S)=`"Surface Energy"/"area"`

Or Surface Energy=S x area

=S x 2dl

Therefore S x 2dl =F x d

`"S" = "F"/"2d"`

Surface tension is the surface energy per unit area of the liquid surface.

It can be also defined as Force per unit length on the liquid surface.

Important: At any interface (it is a line that separates two different medium) the surface tension always acts in the equal and opposite direction and it is always perpendicular to the line at the interface.
a) A molecule inside a liquid. Forces on a molecule due to others are shown. The direction of arrows indicates the attraction of repulsion. (b) Same, for a molecule at a surface. (c) Balance of attractive (A) and repulsive (R) forces.
Surface tension and Surface energy: practical applications

Consider a molecule that is present completely inside the liquid and if it is strongly attracted by the neighboring molecules then the surface energy is less.

Consider a molecule that is present partially inside the liquid the force of attraction by the neighboring molecules is lesser as a result surface energy is more.

Consider a molecule whose very little part is inside the water so very small force of attraction by the neighboring molecules as a result of more surface energy.

Conclusion:  A fluid will stick to a solid surface if the surface energy between fluid and solid is smaller than the sum of energies between solidair and fluidair.
This means S_{sf}( solidfluid) < S_{fa}(fluid air) + S_{sa}(Solid air)
Stretching a film:
(a) A film in equilibrium
(b) The film stretched an extra distance
Why does water stick to glass but Mercury doesn’t?
In case of water and glass, water sticks to glass because the surface energy of water and glass is less than the surface energy between water and air and between glass and air. `"S.E"_(("w""g"))< "S.E"_(("w""a"))+ "S.E"_(("g""a"))`
In case of mercury, surface energy between mercury and glass `"S.E"_(("m""g"))`, surface energy between mercury and air `"S.E"_((ma))` and surface energy between air and glass `"S.E"_(("a""g")). "E"_(("m""g"))>"S.E"_(("m""a"))+"S.E"_(("a""g"))`
How detergents work?

Washing alone with the water can remove some of the dirt but it does not remove the grease stains. This is because water does not wet greasy dirt.

We need detergent which mixes water with dirt to remove it from the clothes.

Detergent molecules look like hairpin shape. When we add detergents to the water one end stick to water and the other end sticks to the dirt.

As a result dirt is getting attracted to the detergent molecules and they get detached from the clothes and they are suspended in the water.

Detergent molecules get attracted to water and when water is removed the dirt also gets removed from the clothes.
Angle of Contact:
 The angle of contact is the angle at which a liquid interface meets a solid surface.

It is denoted by θ.

It is different at interfaces of different pairs of liquids and solids.

For example  Droplet of water on a lotus leaf. The droplet of water(Liquid) is in contact with the solid surface which is a leaf.

This liquid surface makes some angle with the solid surface. This angle is known as angle of contact.
Significance of Angle of Contact

Angle of contact determines whether a liquid will spread on the surface of a solid or it will form droplets on it.

If the Angle of contact is obtuse: then droplet will be formed.

If the Angle of contact is acute: then the water will spread.

Case1: When droplet is formed

Consider we have a solid surface, droplet of water which is liquid and air.

The solid liquid interface denoted by S_{sl}, solid air interface denoted by S_{sa }and liquid air interface denoted by S_{la}.

The angle which S_{sl }makes with S_{la}. It is greater than the 900.

Therefore droplet is formed.
Case 2: When water just spreads

The angle which liquid forms with solid surface is less than 900.
Drops and Bubbles
Why water and bubbles are drops?

Whenever liquid is left to itself it tends to acquire the least possible surface area so that it has the least surface energy so it has the most stability.

Therefore for more stability, they acquire the shape of the sphere, as a sphere has the least possible area.
Distinction between Drop, Cavity, and Bubble:
Drop:  Drop is a spherical structure filled with water:
There is only one interface in the drop.
The interface separates water and air.
Example: Water droplet.
Water droplets:
Cavity: Cavity is a spherical shape filled with air.
In the surroundings there is water and in the middle, there is a cavity filled with air.
There is only one interface which separates air and water.
Example: bubble inside the aquarium.
Bubble: In a bubble there are two interfaces. One is airwater and another is water and air.
Inside a bubble there is air and there is air outside.
But it consists of a thin film of water.
Pressure inside a drop and a cavity

The pressure inside a drop is greater than the pressure outside.

Suppose there is a spherical drop of water of radius ‘r’ which is in equilibrium.

Consider there is an increase in radius which is Δr.
Therefore extra surface energy = surface tension(S) x area
`=S_"l"a xx 4pi(r+Deltar)^2S_"l"a xx 4pir^2`
after calculating
extra surface energy = `8piDeltarS_la`
At equilibrium, extra surface energy = energy gain due to the pressure difference
`8pir Deltar S_la = (P_iP_o)4pir^2xxDeltar`
where `P_i` = pressure inside the drop and `P_o` = pressure outside the drop.
After calculation`P_i  P_o=(2S_(la))/r`
Pressure inside a Bubble

Pressure inside a bubble is greater than the pressure outside.
 As bubble has 2 interfaces, `"P"_i"P"_o=(4"S"_"la")/("r")`
 This is probably why you have to blow hard, but not too hard, to form a soap bubble. A little extra air pressure is needed inside!
Capillary Rise

In Latin the word capilla means hair.

Due to the pressure difference across a curved liquidair interface the water rises up in a narrow tube in spite of gravity.

Consider a vertical capillary tube of circular cross section (radius a) inserted into an open vessel of water.

The contact angle between water and glass is acute. Thus the surface of water in the capillary is concave. As a result there is a pressure difference between the two sides of the top surface. This is given by
`("P"_i"P"_o)=("2S"/r)="2S"/(a sec theta)=("2S"/a) cos theta` ...(i)
Thus the pressure of the water inside the tube, just at the meniscus is less than the atmospheric pressure.
consider the two points A and B. They must be at the same pressure.
`"P"_0+"h"rho"g"="P"_i="P"_"A"` ...(ii)
Where `rho` is the density of water, and h is called the capillary.
By using eq (i) & (ii)
`"h"rho"g" = ("P"_i"P"_o)=(2"S" cos theta)/a`
Therefore the capillary rise is due to surface tension. It is larger, for a smaller radius.
Problem: The lower end of a capillary tube of diameter 2.00 mm is dipped 8.00 cm below the surface of the water in a beaker. What is the pressure required in the tube in order to blow a hemispherical bubble at its end in the water? The surface tension of water at temperature of the experiments is 7.30 ×10^{2} Nm^{1}. 1 atmospheric pressure = 1.01 × 10^{5} Pa, density of water = 1000 kg/m^{3}, g = 9.80 m s^{2}.Also, calculate the excess pressure.
Answer:
The excess pressure in a bubble of gas in a liquid is given by `"2S"/r`, where 'S' is the surface tension of the liquidgas interface. As there is only one liquid surface, therefore using the formula pressure is `"4S"/r`. The radius of the bubble is 'r'. Now the pressure outside the bubble P_{o} equals atmospheric pressure plus the pressure due to 8.00 cm of water column. That is
`"P"_o=(1.01xx105"Pa"+0.08"m"xx1000"kgm"^3xx9.80"ms"^2)`
`= 1.01784 xx 10^5 "Pa"`
Therefore, the pressure inside the bubble is
`"P"_i="P"_o + "2S"/"r"`
`1.01784 xx 10^5Pa +((2xx7.3xx10^2)/10^3 )`
`=(1.01784+0.00146)xx10^5"Pa"`
`=1.02xx10^5"Pa"` where the radius of the bubble is taken to be equal to the radius of the capillary tube since the bubble is hemispherical.